{"product_id":"berryphase-geometric-phase","title":"Berryphase – The Berry Phase \/ γ = −Ω\/2 Physics Sticker","description":"\u003cp\u003eBerryphase, a scaled dragon-knight, coils a glowing violet helix beside him.\u003c\/p\u003e\u003ch3\u003eThe Physics\u003c\/h3\u003e\u003cp\u003eCarry a quantum system slowly round a closed loop of its settings and it returns to its starting state, apart from a phase. Michael Berry showed in 1984 that part of this phase depends only on the geometry of the loop, not on how long it takes (Proc. R. Soc. A 392, 45). For a spin-½ in the state aligned with a magnetic field (m = +½), with the field direction swept slowly round a closed loop, the geometric phase is minus half the solid angle that loop encloses: γ = −mΩ = −Ω\/2. The anti-aligned state (m = −½) picks up +Ω\/2.\u003c\/p\u003e\u003cp\u003e\u003cstrong\u003eγ = −Ω\/2\u003c\/strong\u003e\u003c\/p\u003e\u003cul\u003e\n\u003cli\u003eγ: geometric (Berry) phase of the aligned state (rad)\u003c\/li\u003e\n\u003cli\u003eΩ: solid angle enclosed by the loop traced by the field direction, seen from the origin (sr)\u003c\/li\u003e\n\u003cli\u003em: spin projection along the field in units of ħ (dimensionless; +½ aligned, −½ anti-aligned)\u003c\/li\u003e\n\u003c\/ul\u003e\u003cp\u003eClassical cousins include the Foucault pendulum's precession and parallel transport on a sphere. Berry phases underlie the quantum Hall effect and topological materials.\u003c\/p\u003e\u003cp\u003e\u003cem\u003eHonest caveat:\u003c\/em\u003e In general a state with spin projection m picks up γ = −mΩ (defined modulo 2π), so the sign flips between aligned and anti-aligned states. The adiabatic limit and a non-degenerate state are assumed.\u003c\/p\u003e\u003ch3\u003eOn the sticker\u003c\/h3\u003e\u003cp\u003eBerryphase stands beside a glowing violet helix of light. Printed text: \u003cstrong\u003eBERRYPHASE\u003c\/strong\u003e with the line \u003cstrong\u003eGAMMA = -OMEGA\/2\u003c\/strong\u003e (γ = −Ω\/2 spelled out: the phase for the spin aligned with the field, m = +½; the anti-aligned state gets +Ω\/2). Pixel-art kiss-cut vinyl sticker, white, in 2″, 3″, 4″ and 6″ sizes.\u003c\/p\u003e","brand":"Printify","offers":[{"title":"2\" × 2\" \/ White","offer_id":50964291354727,"sku":"54848263795857279802","price":4.99,"currency_code":"USD","in_stock":true},{"title":"3\" × 3\" \/ White","offer_id":50964291387495,"sku":"29463589810533524498","price":6.99,"currency_code":"USD","in_stock":true},{"title":"4\" × 4\" \/ White","offer_id":50964291420263,"sku":"30661332122886320016","price":8.99,"currency_code":"USD","in_stock":true},{"title":"6\" × 6\" \/ White","offer_id":50964291453031,"sku":"31516324522545060245","price":11.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0596\/8746\/9159\/files\/5588679383539282541_1200.jpg?v=1793674274","url":"https:\/\/pixelatedphysics.com\/fr\/products\/berryphase-geometric-phase","provider":"Pixelated Physics","version":"1.0","type":"link"}